Hi everyone, and welcome back to the Singapore Maths Academy! Today, we’re going to tackle another challenging question that one of our parents has sent through. I thought it would be beneficial for more than one family to have a look at it because it is a real head-scratcher. So, let’s jump onto our iPad and figure out what this question is all about.
The Question
As you may realise, this is quite a challenging question.
Breaking it Down
They’ve given us the area of a triangle, and the triangle is part of the square, but the question is asking us to work out the perimeter. If we want to work out the perimeter, we need to know the length of the square. So, how do we work out the length of the square? That’s the first thought that comes to mind.
If we want to work out the length of the square, where do we start? Well, they’ve given us the area of the triangle. If they’ve given us the area of a triangle, it’s likely that we can work out the area of the square. And if we’ve got the area of the square, perfect! Because then we can work out the length, and from the length, we can work out the perimeter.
So, there are quite a few steps involved, as you can imagine. But in my mind, the target really is to work out the actual length of this square. Let’s start there.
Midpoints and Equal Lengths
The question states that the triangle is formed by the corner and the midpoints of the two sides of the square. If it’s the midpoints of the two sides of the square, then that must mean that the length from the corner to the midpoint is exactly half of the side length, and this is true for both sides.
Fractional Areas
Now, if the area of the triangle is 150 square centimetres, that’s going to be a fraction of the whole square. We need to work out what fraction of the whole square that is. One way we can do that is to work out what fraction the three triangles (let’s call them a, b, and c) are of the whole square.
If we make a straight line from the corner to the opposite midpoint, we can see that triangle a is a quarter of the whole square. The same is true for triangle b.
Triangle c is a bit more tricky, but we can get there. If we draw lines from the midpoints to the centre of the square, we can see that the square is divided into four quarters, and triangle c is half of one of those quarters. Half of a quarter is one-eighth. So, triangle c represents one-eighth of the square.
Putting it Together
So, all together, the unshaded sections (the three triangles) are going to be two-eighths (a), two-eighths (b), and one-eighth (c). That’s five-eighths in total. Therefore, the shaded triangle represents three-eighths of the square.
We’ve been told in the question that the area of the shaded triangle is 150 square centimetres, and that represents three-eighths of the area of the square.
If three-eighths of the square is 150 square centimetres, then one-eighth of the square is 50 square centimetres (150 ÷ 3 = 50). And if one-eighth is 50 square centimetres, then the whole square (eight-eighths) must be 400 square centimetres (50 × 8 = 400).
The Final Stretch
Now that we know the area of the square is 400 square centimetres, we can work out the length of its sides. We know that the area of a square is the length multiplied by the width, but in a square, the length and width are the same. So, we’re looking for a number that, when multiplied by itself, gives us 400. That number is 20.
So, the length of each side of the square is 20 centimetres. Now that we know the length, we can easily work out the perimeter. The perimeter of a square is just four times the length of one side. In this case, that’s 4 × 20 = 80 centimetres.
And there we have it! The area of the square is 400 square centimetres, and the perimeter is 80 centimetres.
Wrapping Up
This was quite a tricky problem, wouldn’t you agree? There were quite a few steps to go through. If you’re not sure about any of these steps, let me know in the messages below, or if you’re in our telegram group, let us know how you went. Did you solve it by yourself, or did you need a little bit of help from me?
Thank you so much for watching, and I’ll see you in our next video. Take care, everyone. Bye!